A guard observes an enemy boat, from an observation tower at a height of 180 m above sea level, to be at an angle of depression of `29^(@)`
Calculate, to the nearest metre, the distance of the boat from the foot of the observation tower.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a guard observing a boat from a tower that is 180 m high. The angle of depression to the boat is 29 degrees. We need to find the horizontal distance from the base of the tower to the boat.

### Step 2: Draw a Diagram
Draw a right triangle where:
- The vertical side (height of the tower) is 180 m.
- The horizontal side (distance from the foot of the tower to the boat) is what we need to find (let's call it \( x \)).
- The angle of depression from the top of the tower to the boat is 29 degrees.

### Step 3: Relate the Angles
Since the angle of depression from the guard to the boat is 29 degrees, the angle of elevation from the boat to the guard is also 29 degrees (alternate interior angles).

### Step 4: Use Trigonometry
We can use the tangent function, which relates the angle of a right triangle to the opposite side and the adjacent side:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case:
\[ \tan(29^\circ) = \frac{180}{x} \]

### Step 5: Rearrange the Equation
Rearranging the equation to solve for \( x \):
\[ x = \frac{180}{\tan(29^\circ)} \]

### Step 6: Calculate \( \tan(29^\circ) \)
Using a calculator, we find:
\[ \tan(29^\circ) \approx 0.554 \]

### Step 7: Substitute and Calculate \( x \)
Now substitute the value of \( \tan(29^\circ) \) into the equation:
\[ x = \frac{180}{0.554} \approx 324.90 \]

### Step 8: Round to the Nearest Metre
Rounding 324.90 to the nearest metre gives us:
\[ x \approx 325 \text{ m} \]

### Final Answer
The distance of the boat from the foot of the observation tower is approximately **325 metres**.

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